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Theorem ofcc 31041
Description: Left operation by a constant on a mixed operation with a constant. (Contributed by Thierry Arnoux, 31-Jan-2017.)
Hypotheses
Ref Expression
ofcc.1 (𝜑𝐴𝑉)
ofcc.2 (𝜑𝐵𝑊)
ofcc.3 (𝜑𝐶𝑋)
Assertion
Ref Expression
ofcc (𝜑 → ((𝐴 × {𝐵})∘𝑓/𝑐𝑅𝐶) = (𝐴 × {(𝐵𝑅𝐶)}))

Proof of Theorem ofcc
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 ofcc.2 . . . 4 (𝜑𝐵𝑊)
2 fnconstg 6393 . . . 4 (𝐵𝑊 → (𝐴 × {𝐵}) Fn 𝐴)
31, 2syl 17 . . 3 (𝜑 → (𝐴 × {𝐵}) Fn 𝐴)
4 ofcc.1 . . 3 (𝜑𝐴𝑉)
5 ofcc.3 . . 3 (𝜑𝐶𝑋)
6 fvconst2g 6789 . . . 4 ((𝐵𝑊𝑥𝐴) → ((𝐴 × {𝐵})‘𝑥) = 𝐵)
71, 6sylan 572 . . 3 ((𝜑𝑥𝐴) → ((𝐴 × {𝐵})‘𝑥) = 𝐵)
83, 4, 5, 7ofcfval 31033 . 2 (𝜑 → ((𝐴 × {𝐵})∘𝑓/𝑐𝑅𝐶) = (𝑥𝐴 ↦ (𝐵𝑅𝐶)))
9 fconstmpt 5460 . 2 (𝐴 × {(𝐵𝑅𝐶)}) = (𝑥𝐴 ↦ (𝐵𝑅𝐶))
108, 9syl6eqr 2825 1 (𝜑 → ((𝐴 × {𝐵})∘𝑓/𝑐𝑅𝐶) = (𝐴 × {(𝐵𝑅𝐶)}))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1508  wcel 2051  {csn 4435  cmpt 5004   × cxp 5401   Fn wfn 6180  cfv 6185  (class class class)co 6974  𝑓/𝑐cofc 31030
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-13 2302  ax-ext 2743  ax-rep 5045  ax-sep 5056  ax-nul 5063  ax-pr 5182
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-3an 1071  df-tru 1511  df-ex 1744  df-nf 1748  df-sb 2017  df-mo 2548  df-eu 2585  df-clab 2752  df-cleq 2764  df-clel 2839  df-nfc 2911  df-ne 2961  df-ral 3086  df-rex 3087  df-reu 3088  df-rab 3090  df-v 3410  df-sbc 3675  df-csb 3780  df-dif 3825  df-un 3827  df-in 3829  df-ss 3836  df-nul 4173  df-if 4345  df-sn 4436  df-pr 4438  df-op 4442  df-uni 4709  df-iun 4790  df-br 4926  df-opab 4988  df-mpt 5005  df-id 5308  df-xp 5409  df-rel 5410  df-cnv 5411  df-co 5412  df-dm 5413  df-rn 5414  df-res 5415  df-ima 5416  df-iota 6149  df-fun 6187  df-fn 6188  df-f 6189  df-f1 6190  df-fo 6191  df-f1o 6192  df-fv 6193  df-ov 6977  df-oprab 6978  df-mpo 6979  df-ofc 31031
This theorem is referenced by: (None)
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